An Extremal Property of the Regular Simplex

Summary

If C is a convex body in Rⁿ such that the ellipsoid of minimal volume containing C-the Liiwner ellipsoid-is the euclidean ball B ⁿ ₂, then the mean width of C is no smaller than the mean width of a regular simplex inscribed in B ⁿ ₂.

1. Introduction and Notation

Suppose that C is a convex body in ℝⁿ such that 0 is an interior point of C, then the mean width ω( C) is defined by where Cn is a constant depending only on the dimension, σ a the normalized Haar measure on the sphere s^n-1 and ‘ϒ_n the n-dimensional standard gaussian measure. Denoting by C* the polar of C with respect to 0 and by ||.|| _c the gauge of C, we obtain the well known formula

The euclidean ball B ⁿ ₂ is the Löwner ellipsoid of C if and only if B ⁿ ₂ is the John ellipsoid of C* i.e., the ellipsoid of maximal volume contained in C*. Hence, in order to prove that the regular simplex has minimal mean width, it is enough to prove that for all convex bodies K whose John ellipsoid is the euclidean ball, we necessarily have 𝓁(K) ≥ 𝓁 (T), i.e., the 𝓁 -norm of K is bounded from below by the 𝓁 -norm of the regular simplex T.